Mathematical Physics


Using a Quotient Polynomial to Probe the Solvability of Polynomial Potentials in One-Dimensional Quantum Mechanics

Authors: Spiros Konstantogiannis

Making use of the Bethe ansatz, we introduce a quotient polynomial and we show that the presence of intermediate terms in it, i.e. terms other than the constant and the leading one, constitutes a non-solvability condition of the respective potential. In this context, both the exact solvability of the quantum harmonic oscillator and the quasi-exact solvability of the sextic anharmonic oscillator stem naturally from the quotient polynomial, as in the first case, it is an energy-dependent constant, while in the second case, it is a second-degree binomial with no linear term. In all other cases, the quotient polynomial has at least one intermediate term, the presence of which makes the respective potential non-solvable.

Comments: 17 Pages.

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Submission history

[v1] 2017-08-24 06:50:21
[v2] 2017-10-29 11:13:19

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